We propose a design for a quantum-information processor where qubits are encoded into hyperfine states of ions held in a linear array of individually tailored linear microtraps and sitting in a spatially varying magnetic field. The magnetic field gradient introduces spatially dependent qubit transition frequencies and a type of spin-spin interaction between qubits. Single- and multiqubit manipulation is achieved via resonant microwave pulses as in liquid-NMR quantum computation while the qubit readout and reset is achieved through trappedion fluorescence shelving techniques. By adjusting the microtrap configurations we can tailor, in hardware, the qubit resonance frequencies and coupling strengths. We show that the system possesses a sideband transition structure which does not scale with the size of the processor, allowing scalable frequency discrimination between qubits. By using large magnetic field gradients, one can reset individual qubits in the ion chain via frequency selective optical pulses to implement quantum-error correction, thus avoiding the need for many tightly focused laser beams.